22 July 2002

Physics Letters A 300 (2002) 18–26

www.elsevier.com/locate/pla

Non-Hermitian Hamiltonians with real and complex eigenvaluesin a Lie-algebraic framework

B. Bagchia, C. Quesneb,∗,1

a Department of Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Calcutta 700 009, Indiab Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229,

Boulevard du Triomphe, B-1050 Brussels, Belgium

Received 7 May 2002; accepted 16 May 2002

Communicated by P.R. Holland

Abstract

We show that complex Lie algebras (in particular sl(2,C)) provide us with an elegant method for studying the transition fromreal to complex eigenvalues of a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized Pöschl–Teller, andMorse. The characterizations of these Hamiltonians under the so-called pseudo-Hermiticity are also discussed. 2002 ElsevierScience B.V. All rights reserved.

PACS: 02.20.Sv; 03.65.Fd; 03.65.Ge

Keywords: Non-Hermitian Hamiltonians; PT symmetry; Pseudo-Hermiticity; Lie algebras

1. Introduction

Some years ago, it was suggested [1] that PT symmetry might be responsible for some non-HermitianHamiltonians to preserve the reality of their bound-state eigenvalues provided it is not spontaneously broken,in which case their complex eigenvalues should come in conjugate pairs. Following this, several non-HermitianHamiltonians (including the non-PT-symmetric ones [2–4]) with real or complex spectra have been analyzed usinga variety of techniques, such as perturbation theory, semiclassical estimates, numerical experiments, analyticalarguments, and algebraic methods. Among the latter, one may quote those connected with supersymmetrization[2,5–10], or some generalizations thereof [11], quasi-solvability [3,12–16], and potential algebras [4,17].

Recently, it has been shown that under some rather mild assumptions, the existence of real or complex-conjugatepairs of eigenvalues can be associated with a class of non-Hermitian Hamiltonians distinguished by either their so-called (weak)pseudo-Hermiticity (i.e., such thatηHη−1 = H †, whereη is some (Hermitian) linear automorphism)

* Corresponding author.E-mail addresses: bbagchi@cucc.ernet.in (B. Bagchi), cquesne@ulb.ac.be (C. Quesne).

1 Directeur de recherches FNRS.

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00689-8

B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26 19

or their invariance under some antilinear operator [18,19]. In such a context, pseudo-Hermiticity under imaginaryshift of the coordinate has been identified as the explanation of the occurrence of real or complex-conjugateeigenvalues for some non-PT-symmetric Hamiltonians [20].

In the course of time, there has been a growing interest in determining the critical strengths of the interaction atwhich PT symmetry (or some generalization) becomes spontaneously broken, i.e., they appearregular complex-energy solutions, where by regular we mean eigenfunctions satisfying the asymptotic boundary conditionsψ(±∞) → 0, so that they are normalizable in a generalized sense [18,20–22]. Some analytical results have beenobtained both for PT-symmetric potentials [22–25] and for potentials that are pseudo-Hermitian under imaginaryshift of the coordinate [20].

In the present Letter, we wish to show that complex Lie algebras provide us with an easy and elegant methodfor studying the transition from real to complex eigenvalues, corresponding toregular eigenfunctions, of (PT-symmetric or non-PT-symmetric) pseudo-Hermitian and non-pseudo-Hermitian Hamiltonians.

2. Non-Hermitian Hamiltonians in an sl(2,C) framework

The generatorsJ0, J+, J− of the complex Lie algebra sl(2,C), characterized by the commutation relations

(1)[J0, J±] = ±J±, [J+, J−] = −2J0,

can be realized as differential operators [4]

(2)J0 = −i∂

∂φ, J± = e±iφ

[± ∂

∂x+

(i∂

∂φ∓ 1

2

)F(x) + G(x)

],

depending upon a real variablex and an auxiliary variableφ ∈ [0,2π), provided the two complex-valued functionsF(x) andG(x) in (2) satisfy coupled differential equations

(3)F ′ = 1− F 2, G′ = −FG.

Here a prime denotes derivative with respect to spatial variablex.The solutions of Eq. (3) fall into the following three classes:

I: F(x) = tanh(x − c − iγ ), G(x) = (bR + ibI )sech(x − c − iγ ),

II: F(x) = coth(x − c − iγ ), G(x) = (bR + ibI )cosech(x − c − iγ ),

(4)III: F(x) = ±1, G(x) = (bR + ibI )e∓x,

wherec, bR , bI ∈ R and−π4 � γ < π

4 , thus providing us with three different realizations of sl(2,C). For bI =γ = 0, the latter reduce to corresponding realizations of sl(2,R) so(2,1), for whichJ0 = J

†0 andJ− = J

†+ [26].

The sl(2,C) Casimir operator corresponding to the differential realizations of type (2) can be written as

(5)J 2 ≡ J 20 ∓ J0 − J±J∓ = ∂2

∂x2 −(

∂2

∂φ2 + 1

4

)F ′ + 2i

∂

∂φG′ −G2 − 1

4.

In this Letter, we are going to consider the sl(2,C) irreducible representations spanned by the states

(6)|km〉 = Ψkm(x,φ) = ψkm(x)eimφ

√2π

,

with fixedk, for which

(7)J0|km〉 = m|km〉, J 2|km〉 = k(k − 1)|km〉,

20 B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26

and

(8)k = kR + ikI , m = mR + imI , mR = kR + n, mI = kI ,

wherekR, kI ,mR,mI ∈ R andn ∈ N. The states withm = k or n = 0 satisfy the equationJ−|kk〉 = 0, while thosewith higher values ofm (or n) can be obtained from them by repeated applications ofJ+ and use of the relationJ+|km〉 ∝ |km + 1〉.

When the parameterm is real, i.e.,mI = 0, we can get rid of the auxiliary variableφ by extending the definitionof the pseudo-norm with a multiplicative integral overφ from 0 to 2π . In the casem is complex, i.e.,mI �= 0, asimilar result can be obtained through an appropriate change of the integral overφ. In the former (respectively,latter) case,J0 is a Hermitian (respectively, non-Hermitian) operator.

From the second relation in Eq. (7), it follows that the functionsψkm(x) of Eq. (6) obey the Schrödinger equation

(9)−ψ ′′km + Vmψkm = −

(k − 1

2

)2

ψkm,

where the family of potentialsVm is defined by

(10)Vm =(

1

4− m2

)F ′ + 2mG′ + G2.

Since the irreducible representations of sl(2,C) correspond to a given eigenvalue in Eq. (9) and the correspondingbasis states to various potentialsVm, m = k, k + 1, k + 2, . . . , it is clear that sl(2,C) is a potential algebra for thefamily of potentialsVm (see [26] and references quoted therein).

To the three classes of solutions of Eq. (3), given in Eq. (4), we can now associate three classes of potentials:

(11)

I: Vm =[(bR + ibI )

2 − (mR + imI )2 + 1

4

]sech2 τ

− 2(mR + imI )(bR + ibI )sechτ tanhτ, τ = x − c − iγ ,

(12)

II: Vm =[(bR + ibI )

2 + (mR + imI )2 − 1

4

]cosech2 τ

− 2(mR + imI )(bR + ibI )cosechτ cothτ, τ = x − c − iγ ,

(13)III: Vm = (bR + ibI )2e∓2x ∓ 2(mR + imI )(bR + ibI )e

∓x .

It is worth stressing that in the generic case, such complex potentials are not invariant under PT symmetry.Eq. (9) can also be rewritten as

(14)−ψ(m)′′n + Vmψ(m)

n = E(m)n ψ(m)

n ,

with ψkm(x) = ψ(m)n (x) and

(15)E(m)n = −

(mR + imI − n − 1

2

)2

.

Real (respectively, complex) eigenvalues therefore correspond tomI = 0 (respectively,mI �= 0).To be acceptable solutions of Eq. (14), the functionsψ

(m)n (x) have to be regular, i.e., such thatψ

(m)n (±∞) → 0.

It is straightforward to determine under which conditions there exist acceptable solutions of Eq. (14) withn = 0.The functionsψ(m)

0 (x) are indeed easily obtained by solving the first-order differential equationJ−Ψmm(x,φ) = 0.For the three classes of potentials (11)–(13), the results read

(16)I: ψ(m)0 (x) ∝ (sechτ )mR+imI −1/2 exp

[(bR + ibI )arctan(sinhτ )

],

B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26 21

(17)II: ψ(m)0 (x) ∝

(sinh

τ

2

)bR+ibI−mR−imI+1/2(cosh

τ

2

)−bR−ibI−mR−imI+1/2

,

(18)III: ψ(m)0 (x) ∝ exp

[−

(mR + imI − 1

2

)x − (bR + ibI )e

−x

].

Such functions are regular providedmR > 12 andbR > 0, where the second condition applies only to class III.

In the remainder of this Letter, we shall illustrate the general theory developed in the present section with someselected examples.

3. Complexified Scarf II potential

The potential

(19)V (x) = −V1 sech2 x − iV2 sechx tanhx, V1 > 0, V2 �= 0,

which belongs to class I defined in Eq. (11), is a complexification of the real Scarf II potential [27]. It is not onlyinvariant under PT symmetry but also P-pseudo-Hermitian. Comparison between Eqs. (11) and (19) shows that itcorresponds toc = γ = 0 and

(20)b2R − b2

I − m2R + m2

I + 1

4= −V1,

(21)bRbI − mRmI = 0,

(22)mRbR − mIbI = 0,

(23)2(mRbI + mIbR) = V2,

where we may assumebI �= 0 since otherwise the sl(2,C) generators (2) would reduce to sl(2,R) ones.To be able to apply the results of the previous section, the only thing we have to do is to solve Eqs. (20)–(23)

in order to express the sl(2,C) parametersbR, bI , mR , mI in terms of the potential parametersV1, V2. Eqs. (22)and (23) yield

(24)mR = V2bI

2(b2R + b2

I ), mI = V2bR

2(b2R + b2

I ).

On inserting these results into Eqs. (20) and (21), we get the relations

(25)(b2R − b2

I

)(1+ V 2

2

4(b2R + b2

I )2

)= −V1 − 1

4,

(26)bRbI

(1− V 2

2

4(b2R + b2

I )2

)= 0.

The latter is satisfied if eitherbR = 0 orbR �= 0 andb2R + b2

I = 12|V2|. It now remains to solve Eq. (25) in those two

possible cases.If we choosebR = 0, then Eq. (25) reduces to a quadratic equation forb2

I , which has real positive solutions

(27)b2I = 1

4

(√V1 + 1

4+ V2 + εI

√V1 + 1

4− V2

)2

, εI = ±1,

provided|V2| � V1 + 14. Eq. (27) then yields forbI the possible solutions

(28)bI = 1

2ε′I

(√V1 + 1

4+ V2 + εI

√V1 + 1

4− V2

), εI , ε

′I = ±1,

22 B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26

while Eq. (24) leads tomR = V2/(2bI ) andmI = 0.From the regularity conditionmR > 1

2 of ψ(m)0 (x), given in Eq. (16), it then follows thatbI must have the

same sign asV2, which we denote byν. Furthermore, we must chooseε′I = +1 or ε′

I = −εI according to whetherν = +1 or ν = −1.

The first set of solutions of Eqs. (20)–(23), compatible with the regularity condition ofψ(m)0 (x), is therefore

given by

bR = 0, bI = 1

2ν

(√V1 + 1

4+ |V2| − ε

√V1 + 1

4− |V2|

),

(29)mR = 1

2

(√V1 + 1

4+ |V2| + ε

√V1 + 1

4− |V2|

), mI = 0, ε = ±1,

whereε = −εI , provided|V2| � V1 + 14 and

√V1 + 1

4 + |V2| + ε

√V1 + 1

4 − |V2| > 1.On inserting these results into Eq. (15), we get two series of real eigenvalues

(30)En,ε = −[

1

2

(√V1 + 1

4+ |V2| + ε

√V1 + 1

4− |V2|

)− n− 1

2

]2

, ε = ±1.

By studying the regularity condition of the associated eigenfunctions obtained by successive applications ofJ+ on

ψ(m)0 (x), it can be shown thatn is restricted to rangen = 0,1,2, . . . < 1

2

(√V1 + 1

4 + |V2|+ ε

√V1 + 1

4 − |V2|−1).

If, on the contrary, we choosebR �= 0 andb2R + b2

I = 12|V2|, then Eq. (25) leads tob2

R − b2I = −1

2(V1 + 14), so

that

(31)bR = 1

2εR

√|V2| − V1 − 1

4, bI = 1

2εI

√|V2| + V1 + 1

4, εR, εI = ±1,

provided|V2| >V1 + 14.

On inserting such results into Eq. (24) and imposing the regularity conditionmR > 12, we obtainε = ν. The

second set of solutions of Eqs. (20)–(23), compatible with the regularity condition ofψ(m)0 (x), is therefore given by

bR = 1

2νε

√|V2| − V1 − 1

4, bI = 1

2ν

√|V2| + V1 + 1

4,

(32)mR = 1

2

√|V2| + V1 + 1

4, mI = 1

2ε

√|V2| − V1 − 1

4, ε = ±1,

where we have setε = νεR . Here we must assume|V2| >V1 + 14 and|V2| + V1 + 1

4 > 1.This set of solutions is associated with a series of complex-conjugate pairs of eigenvalues

(33)En,ε = −[

1

2

(√|V2| + V1 + 1

4+ iε

√|V2| − V1 − 1

4

)− n − 1

2

]2

, ε = ±1,

where it can be shown thatn varies in the rangen = 0,1,2, . . . < 12

(√|V2| + V1 + 14 − 1

).

We conclude that for increasing values of|V2|, the two series of real eigenvalues (30) merge when|V2| reachesthe valueV1 + 1

4, then disappear while complex-conjugate pairs of eigenvalues (33) make their appearance, asalready found elsewhere by another method [22]. Had we chosen the parametrizationV1 = B2 + A(A + 1),V2 = −B(2A + 1), with A andB real, as we did in Ref. [4], we would obtain that the condition|V2| � V1 + 1

4 isalways satisfied, thus only getting the two series of real eigenvalues (30).

B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26 23

4. Complexified generalized Pöschl–Teller potential

We next consider the complexification of the generalized Pöschl–Teller potential [27], namely

(34)V (x) = V1 cosech2 τ − V2 cosechτ cothτ, τ = x − c − iγ , V1 > −1

4, V2 �= 0.

It is easy to recognize (34) to belong to class II defined in Eq. (12). Note that the above potential is PT-symmetricas well as P-pseudo-Hermitian. Comparing with (12), we get

(35)b2R − b2

I + m2R − m2

I − 1

4= V1,

(36)bRbI + mRmI = 0,

(37)2(mRbR − mIbI ) = V2,

(38)mRbI + mIbR = 0.

This time there is no reason to assume thatbI �= 0, since the presence ofγ �= 0 in the generators (2) ensures thatwe are dealing with sl(2,C).

On successively considering the cases wherebI = 0 orbI �= 0 and proceeding as in the previous section, we areled to the two following sets of solutions of Eqs. (35)–(38):

bR = 1

2ν

(√V1 + 1

4+ |V2| − ε

√V1 + 1

4− |V2|

), bI = 0,

(39)mR = 1

2

(√V1 + 1

4+ |V2| + ε

√V1 + 1

4− |V2|

), mI = 0, ε = ±1,

provided|V2| � V1 + 14 and

√V1 + 1

4 + |V2| + ε

√V1 + 1

4 − |V2| > 1, and

bR = 1

2ν

√|V2| + V1 + 1

4, bI = −1

2νε

√|V2| − V1 − 1

4,

(40)mR = 1

2

√|V2| + V1 + 1

4, mI = 1

2ε

√|V2| − V1 − 1

4, ε = ±1,

provided|V2| >V1 + 14 and|V2| + V1 + 1

4 > 1. In both cases,ν denotes the sign ofV2.Comparison with Eq. (15) shows that the first type solutions (39) lead to two series of real eigenvalues

(41)En,ε = −[

1

2

(√V1 + 1

4+ |V2| + ε

√V1 + 1

4− |V2|

)− n− 1

2

]2

, ε = ±1,

while the second type solutions (40) correspond to a series of complex-conjugate pairs of eigenvalues

(42)En,ε = −[

1

2

(√|V2| + V1 + 1

4+ iε

√|V2| − V1 − 1

4

)− n − 1

2

]2

, ε = ±1.

In the former (respectively, latter) case, it can be shown thatn varies in the rangen = 0,1,2, . . . <12

(√V1 + 1

4 + |V2| + ε

√V1 + 1

4 − |V2| − 1)

(respectively,n = 0,1,2, . . . < 12

(√|V2| + V1 + 14 − 1

)).

For increasing values of|V2|, we observe a phenomenon entirely similar to that already noted for thecomplexified Scarf II potential: disappearance of the real eigenvalues and simultaneous appearance of complex-conjugate ones at the threshold|V2| = V1 + 1

4. In this case, however, only partial results were reported inthe literature. In Ref. [4], we obtained the two series of real eigenvalues (41) using the parametrizationV1 =B2 +A(A+ 1), V2 = B(2A+ 1), with A andB real, so that the condition|V2| � V1 + 1

4 is automatically satisfied.

24 B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26

Furthermore, Lévai and Znojil considered both the real [8] and the complex [24] eigenvalues in a parametrizationV1 = 1

4[2(α2 +β2)−1], V2 = 12(β

2 −α2), whereinα andβ are real or one of them is real and the other imaginary,respectively. Their results, however, disagree with ours in both cases.

5. Complexified Morse potential

The potential

(43)V (x) = (V1R + iV1I )e−2x − (V2R + iV2I )e

−x, V1R,V1I ,V2R,V2I ∈ R,

is the most general potential of class III for the upper sign choice in Eq. (13) and is a complexification of thestandard Morse potential [27]. Comparison with Eq. (13) shows that

(44)b2R − b2

I = V1R,

(45)2bRbI = V1I ,

(46)2(mRbR − mIbI ) = V2R,

(47)2(mRbI + mIbR) = V2I ,

where we may assumebI �= 0.On solving Eq. (45) forbR and inserting the result into Eq. (44), we get a quadratic equation forb2

I , of whichwe only keep the real positive solutions. The results forbR andbI read

(48)bR = 1√2εI ν(V1R + ∆)1/2, bI = 1√

2εI (−V1R + ∆)1/2, ∆ =

√V 2

1R + V 21I , εI = ±1,

whereV1I �= 0 if V1R � 0 andν denotes the sign ofV1I . On introducing Eq. (48) into Eqs. (46) and (47) andsolving formR andmI , we then obtain

(49)mR = εI ν

2√

2∆

[(V1R + ∆)1/2V2R + ν(−V1R + ∆)1/2V2I

],

(50)mI = εI ν

2√

2∆

[(V1R + ∆)1/2V2I − ν(−V1R + ∆)1/2V2R

].

From the regularity conditionsbR > 0 andmR > 12 of ψ(m)

0 (x), given in Eq. (18), it follows that we must chooseεI = ν, V1I �= 0 if V1R < 0, and

(51)(V1R + ∆)1/2V2R + ν(−V1R + ∆)1/2V2I >√

2∆.

We conclude thatV1I �= 0 must hold for any value ofV1R.Real eigenvalues are associated withmI = 0 and therefore occur whenever the condition

(52)(V1R + ∆)1/2V2I = ν(−V1R + ∆)1/2V2R

is satisfied. In such a case,V2I can be expressed in terms ofV1R, V1I , andV2R, so that the real eigenvalues aregiven by

(53)En = −[

V2R√2|V1I |

(−V1R + ∆)1/2 − n− 1

2

]2

.

It can be shown that regular eigenfunctions correspond ton = 0,1,2, . . . < (V2R/√

2|V1I |)(−V1R + ∆)1/2 − 12.

B. Bagchi, C. Quesne / Physics Letters A 300 (2002) 18–26 25

Furthermore, when condition (52) is not fulfilled but condition (51) holds, we get complex eigenvaluesassociated with regular eigenfunctions,

(54)En = −{

1

2√

2∆

[(V1R + ∆)1/2 − iν(−V1R + ∆)1/2](V2R + iV2I ) − n − 1

2

}2

,

wheren = 0,1,2, . . . < 12√

2∆[(V1R + ∆)1/2V2R + ν(−V1R + ∆)1/2V2I ] − 1

2.

It should be stressed that contrary to what happens for the two previous examples, here the real eigenvalues,belonging to a single series, only occur for a special value of the parameterV2I , while the complex eigenvalues,which do not appear in complex-conjugate pairs (sinceE∗

n corresponds toV ∗(x)), are obtained for generic valuesof V2I .

To interpret such results, it is worth choosing the parametrizationV1R = A2 − B2, V1I = 2AB, V2R = γA,V2I = δB, whereA, B, γ , δ are real,A > 0, andB �= 0. The complexified Morse potential (43) can then beexpressed as

(55)V (x) = (A + iB)2e−2x − (2C + 1)(A+ iB)e−x, C = (γ − 1)A + i(δ − 1)B

2(A + iB).

Its (real or complex) eigenvalues can be written in a unified way asEn = −(C − n)2, while the regularitycondition (51) amounts to(γ − 1)A2 + (δ − 1)B2 > 0.

For δ = γ > 1, and thereforeC = 12(γ − 1) ∈ R+, the potential (55) coincides with that considered in our

previous work [4]. Such a potential was shown to be pseudo-Hermitian under imaginary shift of the coordinate [20].We confirm here that it has only real eigenvalues corresponding ton = 0,1,2, . . . < C, thus exhibiting no symmetrybreaking over the whole parameter range. For the values ofδ different fromγ , the potential indeed fails to bepseudo-Hermitian. In such a case,C is complex as well as the eigenvalues. The eigenfunctions associated withn = 0,1,2, . . . < ReC are however regular. The existence of regular eigenfunctions with complex energies forgeneral complex potentials is a phenomenon that has been known for some time (see, e.g., [28]).

6. Conclusion

In the present Letter, we have shown that complex Lie algebras (in particular sl(2,C)) provide us with anelegant tool to easily determine both real and complex eigenvalues of non-Hermitian Hamiltonians, correspondingto regular eigenfunctions. For such a purpose, it has been essential to extend the scope of our previous work [4] tothose Lie algebra irreducible representations that remain non-unitary in the real algebra limit (namely those withkI �= 0).

We have illustrated our method by deriving the real and complex eigenvalues of the PT-symmetric complexifiedScarf II potential, previously determined by other means [22]. In addition, we have established similar results forthe PT-symmetric generalized Pöschl–Teller potential, for which only partial results were available [4,8,24]. Wehave shown that in both cases symmetry breaking occurs for a given value of one of the potential parameters.

Finally, we have considered a more general form of the complexified Morse potential than that previouslystudied [4,19,20]. For a special value of one of its parameters, our potential reduces to the former one and becomespseudo-Hermitian under imaginary shift of the coordinate. We have proved that here no symmetry breaking occurs,the complex eigenvalues being associated with non-pseudo-Hermitian Hamiltonians.

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